#1      2x -  y +  z =  7
#2      3x + 2y - 2z = -7
#3       x - 3y + 4z = 20

All three equations contain all three variables, so

1. Pick a variable to eliminate.  

I'll pick x.

2. Pick two of the equations to eliminate that letter from.

I'll pick #1 & #3

3. Eliminate that variable from them

#1      2x -  y +  z =  7
#3       x - 3y + 4z = 20

Multiply #2 through by -2 and add them:

        2x -  y +  z =   7 
       -2x + 6y - 8z = -40
        ------------------
             5y - 7z = -33 

4. Pick either one of those equations

I'll pick #3

5. Put it together with the equation you haven't yet used.

I haven't used #2 yet, so I'll put #2 and #3 together:

#2      3x + 2y - 2z = -7
#3       x - 3y + 4z = 20

6. Eliminate that same variable from them 

Multiply #3 through by -3 and add them:

        3x + 2y +  2z =   9 
       -3x + 9y - 12z = -60
        -------------------
            11y - 10z = -51

7.  Put the two equations with the eliminated variable 
    together as a system of two equations in two variables:

             5y -  7z = -33
            11y - 10z = -51 

8.  Solve them for the two variables that they contain.

Eliminate z by mutiplying the first one through by 10,
and the second one through by -7

            50y - 70z = -330
           -77y + 70z =  357
           -----------------
           -27y       =   27
              y       =   -1

Substitute -1 for y into:    

             5y -  7z = -33
           5(-1) - 7z = -33
             -5  - 7z = -33
                  -7z = -28
                    z = 4

9.  Pick one of the original equations to substitute
    the two variables you have found in the preceding step.

    I'll pick #3

#3       x - 3y + 4z = 20

10. Substitute and solve for the remaining variable (the first one
    picked in step 1.

    I'll substitute y = -1 and z = 4 into #3

         x - 3y + 4z = 20
    x - 3(-1) + 4(4) = 20
          x + 3 + 16 = 20
              x + 19 = 20
                   x = 1

So the solution is (x,y,z) = (1,-1,4)

Edwin